Before we present Lemmas REF  and REF , we first state Lemma REF  below, which is the tool we use in the proofs of Lemmas REF  and REF  (see the flowchart in Figure REF ). Lemma REF  is a well-known result that bounds tail probabilities and moments using drift conditions [1]}, [2]}, [3]}. We include it here for completeness, and the form below slightly generalizes the commonly used form in the literature.
[Bounds via Drift]
Let \(\lbrace S(t),t\ge 0\rbrace \)  be a continuous-time Markov chain on a countable state space \(\mathcal {S}\)  and \(r_{s,s^{\prime }}\)  be its transition rate from state \(s\)  to state \(s^{\prime }\) . Assume that it has a unique stationary distribution and let \(S(\infty )\)  be a random element that follows the stationary distribution. Let \(V\colon \mathcal {S}\rightarrow \mathbb {R}_+\)  be a Lyapunov function. Suppose
\(v_{\max }:=\sup _{s,s^{\prime }\colon r_{s,s^{\prime }}>0}|V(s^{\prime })-V(s)|<+\infty ,\qquad \sup _{s}\sum _{s^{\prime }}r_{s,s^{\prime }}<+\infty ,\) 
